The Clausal Theory of Types

Wolfram, D. A.

doi:10.1017/cbo9780511569906

Cited by 63 publications

(15 citation statements)

References 3 publications

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“…In the first case, we find the works on CRS [20] and other higher-order rewriting systems [30,23], in the second case the works on combination of λ-calculus with term rewriting [2,7,12,17] to mention only a few.…”

Section: Introductionmentioning

confidence: 99%

Pure patterns type systems

et al. 2003

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We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with built-in matching facilities. This framework, that we call "Pure Pattern Type Systems", is particularly well-suited for the foundations of programming (meta)languages and proof assistants since it provides in a fully unified setting higher-order capabilities and pattern matching ability together with powerful type systems. We prove some standard properties like confluence and subject reduction for the case of a syntactic theory and under a syntactical restriction over the shape of patterns. We also conjecture the strong normalization of typable terms. This work should be seen as a contribution to a formal connection between logics and rewriting, and a step towards new proof engines based on the Curry-Howard isomorphism.

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Section: Introductionmentioning

confidence: 99%

Pure patterns type systems

et al. 2003

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“…There were proposed several partial solutions of the problem (second order matching- [GH78]; correct, but without a proof of completeness, algorithm- [Wo189]; third order matching- [Dow93]; fourth order matching- [Pad96]). …”

Section: Prefacementioning

confidence: 99%

Linear interpolation for the higher-order matching problem

Schubert¹

1997

TAPSOFT '97: Theory and Practice of Software Development

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Abstract. We present here a particular case of the higher order matching problem --the linear interpolation problem. The problem consists in solving a collection of higher order matching equations of the shape xM1... Mk = N, where x is the only unknown quantity. We prove recursive equivalence of the higher order matching problem and the linear interpolation problem. We also investigate decidability of a special case of the fifth order linear interpolation problem. The restriction we consider consists in that arguments of variables from the main abstraction in terms M1,. 9 9 Mk cannot contain variables from the main abstraction. In this paper, we present the linear interpolation problem. This problem is interesting since to construct a solution for such a problem we deal with a single object, not a set of objects as in the case of the matching problem in general formulation. Moreover, V. Padovani investigates a similar problem in his paper [Pad96]. The Padovani's problem consists in solving the pair of sets {~, gr} of interpolation equations. A solution of such a problem is a concretisation of unknown quantities which satisfies each equation in the set ~ and does not satisfy any equation in the set ~. Decidability of the problem implies decidability of the matching problem as proven in [Pad96].1 This work has been partly supported by ESPRIT BRA 7232 GENTZEN, and KBN 8 TllC 034 10 grants. 442In the second part of the paper, we look into decidability of a special case of the fifth order linear interpolation problem. The restriction we consider is that arguments of variables from the main abstraction in terms M1, 9 9 Mk cannot contain occurrences of variables from the main abstraction.This issue is interesting, since it gives constructors of proof-checkers and proof-assistants possibility of solving some fifth order matching equations. This paper is organised as follows --in Section 2 we present some basic definitions and define some useful notation, in Section 3 we prove recursive equivalence of the higher-order matching problem and the interpolation problem, and in Section 4 we prove our decidability result.The present paper contains only a sketch of the proof. More details can be found in the technical report [Sch96].

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“…The problem is decidable in the case where the order of the type A is 4 or less [2,4,9,14,22,24], as well as in other special cases [3,13,16,17,18,19]. A terminating algorithm has been proposed [25], but it is not known to be complete. In the words of Huet [8], "This vexing but important problem is thus still open after 30 years of intense investigation.…”

Section: Introduction and Related Workmentioning

confidence: 99%

Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form

Støvring¹

2006

BRICS

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Higher-order matching is a special case of unification of simply-typed lambda-terms: in a matching equation, one of the two sides contains no unification variables. Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem.We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is that solutions must be in long beta-eta normal form.

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The Clausal Theory of Types

Cited by 63 publications

References 3 publications

Pure patterns type systems

Pure patterns type systems

Linear interpolation for the higher-order matching problem

Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form

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